High order Nyström methods for transmission problems for Helmholtz equation
This provides a rigorous theoretical foundation for high-order Nyström methods in computational electromagnetics and acoustics, though it is an incremental improvement over existing discretization techniques.
The paper presents superalgebraically convergent Nyström discretizations for all four Helmholtz boundary operators on smooth closed curves in 2D, achieving stable and convergent solutions for integral equation formulations of the Helmholtz equation.
We present superalgebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón's calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually superalgebraic for smooth solutions.